LMFDB - Embedded newform 324.6.e.f.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,6,Mod(109,324)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("324.109");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	324.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$51.9643576194$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 11x^{2} + 10x + 100 $ x^4 - x^3 + 11x^2 + 10x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$1.85078 - 3.20565i$ of defining polynomial
Character	$\chi$	$=$	324.109
Dual form			324.6.e.f.217.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-28.8141 + 49.9074i) q^{5} +(-78.4422 - 135.866i) q^{7} +O(q^{10})$	$q+(-28.8141 + 49.9074i) q^{5} +(-78.4422 - 135.866i) q^{7} +(109.012 + 188.815i) q^{11} +(293.769 - 508.822i) q^{13} +2124.56 q^{17} +29.7687 q^{19} +(-1585.74 + 2746.59i) q^{23} +(-98.0000 - 169.741i) q^{25} +(-1821.07 - 3154.18i) q^{29} +(-3165.94 + 5483.57i) q^{31} +9040.95 q^{35} -8435.69 q^{37} +(-5073.22 + 8787.07i) q^{41} +(3427.88 + 5937.26i) q^{43} +(-12223.9 - 21172.4i) q^{47} +(-3902.85 + 6759.94i) q^{49} -25483.2 q^{53} -12564.4 q^{55} +(10348.1 - 17923.4i) q^{59} +(-22790.1 - 39473.7i) q^{61} +(16929.3 + 29322.5i) q^{65} +(-9549.67 + 16540.5i) q^{67} +23055.0 q^{71} -89651.6 q^{73} +(17102.4 - 29622.1i) q^{77} +(17011.8 + 29465.3i) q^{79} +(54248.7 + 93961.6i) q^{83} +(-61217.3 + 106031. i) q^{85} -110055. q^{89} -92175.4 q^{91} +(-857.757 + 1485.68i) q^{95} +(14416.0 + 24969.2i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{7}+O(q^{10}) $ 4 * q + 32 * q^7	$ 4 q + 32 q^{7} - 486 q^{11} - 208 q^{13} + 3888 q^{17} - 1264 q^{19} - 6804 q^{23} - 392 q^{25} - 11664 q^{29} - 3328 q^{31} + 39852 q^{35} - 19912 q^{37} - 13608 q^{41} - 4960 q^{43} - 18468 q^{47} - 26676 q^{49} - 23328 q^{53} - 106272 q^{55} - 1944 q^{59} - 8176 q^{61} + 79704 q^{65} - 90064 q^{67} - 89424 q^{71} - 242428 q^{73} + 167184 q^{77} - 28768 q^{79} + 15066 q^{83} - 132840 q^{85} - 357696 q^{89} - 484880 q^{91} - 39852 q^{95} - 88942 q^{97}+O(q^{100}) $ 4 * q + 32 * q^7 - 486 * q^11 - 208 * q^13 + 3888 * q^17 - 1264 * q^19 - 6804 * q^23 - 392 * q^25 - 11664 * q^29 - 3328 * q^31 + 39852 * q^35 - 19912 * q^37 - 13608 * q^41 - 4960 * q^43 - 18468 * q^47 - 26676 * q^49 - 23328 * q^53 - 106272 * q^55 - 1944 * q^59 - 8176 * q^61 + 79704 * q^65 - 90064 * q^67 - 89424 * q^71 - 242428 * q^73 + 167184 * q^77 - 28768 * q^79 + 15066 * q^83 - 132840 * q^85 - 357696 * q^89 - 484880 * q^91 - 39852 * q^95 - 88942 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−28.8141	+	49.9074i	−0.515442	+	0.892771i	0.484398	+	0.874848i	$0.339039\pi$
$5$	−28.8141	+	49.9074i	−0.515442	+	0.892771i	−0.999839	+	0.0179231i	$0.994295\pi$
$6$		0			0
$7$	−78.4422	−	135.866i	−0.605068	−	1.04801i	−0.992041	−	0.125918i	$-0.959812\pi$
$7$	−78.4422	−	135.866i	−0.605068	−	1.04801i	0.386972	−	0.922091i	$-0.373521\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	109.012	+	188.815i	0.271640	+	0.470495i	0.969282	−	0.245952i	$-0.0791005\pi$
$11$	109.012	+	188.815i	0.271640	+	0.470495i	−0.697642	+	0.716447i	$0.745767\pi$
$12$		0			0
$13$	293.769	−	508.822i	0.482111	−	0.835041i	−0.517678	−	0.855576i	$-0.673203\pi$
$13$	293.769	−	508.822i	0.482111	−	0.835041i	0.999789	+	0.0205346i	$0.00653681\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2124.56			1.78298			0.891491	−	0.453038i	$-0.149660\pi$
$17$	2124.56			1.78298			0.891491	+	0.453038i	$0.149660\pi$
$18$		0			0
$19$	29.7687			0.0189180			0.00945902	−	0.999955i	$-0.496989\pi$
$19$	29.7687			0.0189180			0.00945902	+	0.999955i	$0.496989\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1585.74	+	2746.59i	−0.625048	+	1.08262i	0.363483	+	0.931601i	$0.381587\pi$
$23$	−1585.74	+	2746.59i	−0.625048	+	1.08262i	−0.988532	+	0.151014i	$0.951746\pi$
$24$		0			0
$25$	−98.0000	−	169.741i	−0.0313600	−	0.0543171i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1821.07	−	3154.18i	−0.402097	−	0.696452i	0.591882	−	0.806025i	$-0.298385\pi$
$29$	−1821.07	−	3154.18i	−0.402097	−	0.696452i	−0.993979	+	0.109573i	$0.965052\pi$
$30$		0			0
$31$	−3165.94	+	5483.57i	−0.591696	+	1.02485i	0.402309	+	0.915504i	$0.368208\pi$
$31$	−3165.94	+	5483.57i	−0.591696	+	1.02485i	−0.994004	+	0.109343i	$0.965125\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9040.95			1.24751
$36$		0			0
$37$	−8435.69			−1.01302			−0.506508	−	0.862235i	$-0.669064\pi$
$37$	−8435.69			−1.01302			−0.506508	+	0.862235i	$0.669064\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5073.22	+	8787.07i	−0.471328	+	0.816365i	−0.999462	−	0.0327965i	$-0.989559\pi$
$41$	−5073.22	+	8787.07i	−0.471328	+	0.816365i	0.528134	+	0.849161i	$0.322892\pi$
$42$		0			0
$43$	3427.88	+	5937.26i	0.282718	+	0.489683i	0.972053	−	0.234760i	$-0.0754305\pi$
$43$	3427.88	+	5937.26i	0.282718	+	0.489683i	−0.689335	+	0.724443i	$0.742097\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−12223.9	−	21172.4i	−0.807171	−	1.39806i	−0.914816	−	0.403872i	$-0.867664\pi$
$47$	−12223.9	−	21172.4i	−0.807171	−	1.39806i	0.107645	−	0.994189i	$-0.465669\pi$
$48$		0			0
$49$	−3902.85	+	6759.94i	−0.232216	+	0.402209i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−25483.2			−1.24613			−0.623066	−	0.782169i	$-0.714113\pi$
$53$	−25483.2			−1.24613			−0.623066	+	0.782169i	$0.714113\pi$
$54$		0			0
$55$	−12564.4			−0.560059
$56$		0			0
$57$		0			0
$58$		0			0
$59$	10348.1	−	17923.4i	0.387017	−	0.670333i	−0.605030	−	0.796203i	$-0.706839\pi$
$59$	10348.1	−	17923.4i	0.387017	−	0.670333i	0.992047	+	0.125870i	$0.0401721\pi$
$60$		0			0
$61$	−22790.1	−	39473.7i	−0.784191	−	1.35826i	−0.929481	−	0.368870i	$-0.879745\pi$
$61$	−22790.1	−	39473.7i	−0.784191	−	1.35826i	0.145290	−	0.989389i	$-0.453589\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	16929.3	+	29322.5i	0.497000	+	0.860830i
$66$		0			0
$67$	−9549.67	+	16540.5i	−0.259897	+	0.450155i	−0.966214	−	0.257741i	$-0.917022\pi$
$67$	−9549.67	+	16540.5i	−0.259897	+	0.450155i	0.706317	+	0.707896i	$0.250355\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	23055.0			0.542773			0.271387	−	0.962470i	$-0.412518\pi$
$71$	23055.0			0.542773			0.271387	+	0.962470i	$0.412518\pi$
$72$		0			0
$73$	−89651.6			−1.96902			−0.984511	−	0.175320i	$-0.943904\pi$
$73$	−89651.6			−1.96902			−0.984511	+	0.175320i	$0.943904\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	17102.4	−	29622.1i	0.328722	−	0.569364i
$78$		0			0
$79$	17011.8	+	29465.3i	0.306678	+	0.531182i	0.977634	−	0.210316i	$-0.0674492\pi$
$79$	17011.8	+	29465.3i	0.306678	+	0.531182i	−0.670955	+	0.741498i	$0.734116\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	54248.7	+	93961.6i	0.864359	+	1.49711i	0.867682	+	0.497120i	$0.165609\pi$
$83$	54248.7	+	93961.6i	0.864359	+	1.49711i	−0.00332246	+	0.999994i	$0.501058\pi$
$84$		0			0
$85$	−61217.3	+	106031.i	−0.919023	+	1.59180i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−110055.			−1.47277			−0.736384	−	0.676564i	$-0.763468\pi$
$89$	−110055.			−1.47277			−0.736384	+	0.676564i	$0.763468\pi$
$90$		0			0
$91$	−92175.4			−1.16684
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−857.757	+	1485.68i	−0.00975114	+	0.0168895i
$96$		0			0
$97$	14416.0	+	24969.2i	0.155566	+	0.269448i	0.933265	−	0.359189i	$-0.116947\pi$
$97$	14416.0	+	24969.2i	0.155566	+	0.269448i	−0.777699	+	0.628637i	$0.783613\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	20145.2	+	34892.5i	0.196502	+	0.340352i	0.947392	−	0.320076i	$-0.103708\pi$
$101$	20145.2	+	34892.5i	0.196502	+	0.340352i	−0.750890	+	0.660428i	$0.770375\pi$
$102$		0			0
$103$	−102274.	+	177143.i	−0.949885	+	1.64525i	−0.204223	+	0.978924i	$0.565467\pi$
$103$	−102274.	+	177143.i	−0.949885	+	1.64525i	−0.745662	+	0.666324i	$0.767867\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	164042.			1.38515			0.692573	−	0.721348i	$-0.256477\pi$
$107$	164042.			1.38515			0.692573	+	0.721348i	$0.256477\pi$
$108$		0			0
$109$	−131554.			−1.06057			−0.530285	−	0.847819i	$-0.677915\pi$
$109$	−131554.			−1.06057			−0.530285	+	0.847819i	$0.677915\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	47161.0	−	81685.2i	0.347446	−	0.601793i	−0.638349	−	0.769747i	$-0.720382\pi$
$113$	47161.0	−	81685.2i	0.347446	−	0.601793i	0.985795	+	0.167953i	$0.0537158\pi$
$114$		0			0
$115$	−91383.4	−	158281.i	−0.644352	−	1.11605i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−166655.	−	288655.i	−1.07883	−	1.86858i
$120$		0			0
$121$	56758.1	−	98307.8i	0.352423	−	0.610414i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−168793.			−0.966226
$126$		0			0
$127$	−303144.			−1.66778			−0.833891	−	0.551929i	$-0.813892\pi$
$127$	−303144.			−1.66778			−0.833891	+	0.551929i	$0.813892\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−158286.	+	274160.i	−0.805871	+	1.39581i	0.109830	+	0.993950i	$0.464969\pi$
$131$	−158286.	+	274160.i	−0.805871	+	1.39581i	−0.915701	+	0.401859i	$0.868364\pi$
$132$		0			0
$133$	−2335.12	−	4044.55i	−0.0114467	−	0.0198263i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	491.479	+	851.266i	0.00223719	+	0.00387493i	0.867142	−	0.498061i	$-0.165955\pi$
$137$	491.479	+	851.266i	0.00223719	+	0.00387493i	−0.864905	+	0.501936i	$0.832621\pi$
$138$		0			0
$139$	−38766.9	+	67146.3i	−0.170186	+	0.294771i	−0.938485	−	0.345320i	$-0.887770\pi$
$139$	−38766.9	+	67146.3i	−0.170186	+	0.294771i	0.768299	+	0.640091i	$0.221104\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	128098.			0.523844
$144$		0			0
$145$	209889.			0.829030
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−59604.6	+	103238.i	−0.219945	+	0.380956i	−0.954791	−	0.297278i	$-0.903921\pi$
$149$	−59604.6	+	103238.i	−0.219945	+	0.380956i	0.734846	+	0.678234i	$0.237255\pi$
$150$		0			0
$151$	29784.4	+	51588.1i	0.106303	+	0.184123i	0.914270	−	0.405105i	$-0.132765\pi$
$151$	29784.4	+	51588.1i	0.106303	+	0.184123i	−0.807967	+	0.589228i	$0.799432\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−182447.	−	316008.i	−0.609969	−	1.05650i
$156$		0			0
$157$	−148781.	+	257696.i	−0.481723	+	0.834368i	−0.999780	−	0.0209776i	$-0.993322\pi$
$157$	−148781.	+	257696.i	−0.481723	+	0.834368i	0.518057	+	0.855346i	$0.326655\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	497557.			1.51279
$162$		0			0
$163$	−98681.9			−0.290917			−0.145458	−	0.989364i	$-0.546466\pi$
$163$	−98681.9			−0.290917			−0.145458	+	0.989364i	$0.546466\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9276.51	−	16067.4i	0.0257391	−	0.0445814i	−0.852869	−	0.522125i	$-0.825139\pi$
$167$	9276.51	−	16067.4i	0.0257391	−	0.0445814i	0.878608	+	0.477544i	$0.158473\pi$
$168$		0			0
$169$	13046.4	+	22597.0i	0.0351377	+	0.0608603i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	248326.	+	430113.i	0.630822	+	1.09262i	0.987384	+	0.158345i	$0.0506157\pi$
$173$	248326.	+	430113.i	0.630822	+	1.09262i	−0.356561	+	0.934272i	$0.616051\pi$
$174$		0			0
$175$	−15374.7	+	26629.7i	−0.0379499	+	0.0657311i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−150072.			−0.350080			−0.175040	−	0.984561i	$-0.556005\pi$
$179$	−150072.			−0.350080			−0.175040	+	0.984561i	$0.556005\pi$
$180$		0			0
$181$	−73515.6			−0.166795			−0.0833976	−	0.996516i	$-0.526577\pi$
$181$	−73515.6			−0.166795			−0.0833976	+	0.996516i	$0.526577\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	243066.	−	421003.i	0.522150	−	0.904391i
$186$		0			0
$187$	231604.	+	401150.i	0.484330	+	0.838885i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	299659.	+	519025.i	0.594353	+	1.02945i	0.993638	+	0.112622i	$0.0359251\pi$
$191$	299659.	+	519025.i	0.594353	+	1.02945i	−0.399285	+	0.916827i	$0.630742\pi$
$192$		0			0
$193$	361873.	−	626783.i	0.699299	−	1.21122i	−0.269411	−	0.963025i	$-0.586829\pi$
$193$	361873.	−	626783.i	0.699299	−	1.21122i	0.968710	−	0.248196i	$-0.0798378\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−416589.			−0.764790			−0.382395	−	0.923999i	$-0.624901\pi$
$197$	−416589.			−0.764790			−0.382395	+	0.923999i	$0.624901\pi$
$198$		0			0
$199$	688177.			1.23188			0.615938	−	0.787794i	$-0.288777\pi$
$199$	688177.			1.23188			0.615938	+	0.787794i	$0.288777\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−285697.	+	494841.i	−0.486592	+	0.842802i
$204$		0			0
$205$	−292360.	−	506382.i	−0.485885	−	0.841577i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3245.16	+	5620.78i	0.00513890	+	0.00890084i
$210$		0			0
$211$	160937.	−	278751.i	0.248857	−	0.431033i	−0.714352	−	0.699787i	$-0.753278\pi$
$211$	160937.	−	278751.i	0.248857	−	0.431033i	0.963209	+	0.268753i	$0.0866117\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−395084.			−0.582899
$216$		0			0
$217$	993373.			1.43207
$218$		0			0
$219$		0			0
$220$		0			0
$221$	624130.	−	1.08102e6i	0.859596	−	1.48886i
$222$		0			0
$223$	177237.	+	306983.i	0.238667	+	0.413383i	0.960332	−	0.278859i	$-0.0899563\pi$
$223$	177237.	+	306983.i	0.238667	+	0.413383i	−0.721665	+	0.692242i	$0.756623\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	113261.	+	196174.i	0.145887	+	0.252684i	0.929704	−	0.368309i	$-0.120063\pi$
$227$	113261.	+	196174.i	0.145887	+	0.252684i	−0.783816	+	0.620993i	$0.786730\pi$
$228$		0			0
$229$	727154.	−	1.25947e6i	0.916299	−	1.58708i	0.111312	−	0.993786i	$-0.464495\pi$
$229$	727154.	−	1.25947e6i	0.916299	−	1.58708i	0.804988	−	0.593292i	$-0.202172\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−423642.			−0.511221			−0.255611	−	0.966780i	$-0.582277\pi$
$233$	−423642.			−0.511221			−0.255611	+	0.966780i	$0.582277\pi$
$234$		0			0
$235$	1.40888e6			1.66420
$236$		0			0
$237$		0			0
$238$		0			0
$239$	373526.	−	646966.i	0.422986	−	0.732633i	−0.573244	−	0.819385i	$-0.694315\pi$
$239$	373526.	−	646966.i	0.422986	−	0.732633i	0.996230	+	0.0867515i	$0.0276486\pi$
$240$		0			0
$241$	−305937.	−	529899.i	−0.339305	−	0.587693i	0.644998	−	0.764185i	$-0.276858\pi$
$241$	−305937.	−	529899.i	−0.339305	−	0.587693i	−0.984302	+	0.176492i	$0.943525\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−224914.	−	389562.i	−0.239387	−	0.414631i
$246$		0			0
$247$	8745.12	−	15147.0i	0.00912059	−	0.0157973i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−985152.			−0.987004			−0.493502	−	0.869745i	$-0.664283\pi$
$251$	−985152.			−0.987004			−0.493502	+	0.869745i	$0.664283\pi$
$252$		0			0
$253$	−691463.			−0.679153
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−693891.	+	1.20185e6i	−0.655327	+	1.13506i	0.326485	+	0.945203i	$0.394136\pi$
$257$	−693891.	+	1.20185e6i	−0.655327	+	1.13506i	−0.981812	+	0.189857i	$0.939197\pi$
$258$		0			0
$259$	661714.	+	1.14612e6i	0.612944	+	1.06165i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−438768.	−	759968.i	−0.391152	−	0.677495i	0.601450	−	0.798911i	$-0.294590\pi$
$263$	−438768.	−	759968.i	−0.391152	−	0.677495i	−0.992602	+	0.121415i	$0.961257\pi$
$264$		0			0
$265$	734274.	−	1.27180e6i	0.642308	−	1.11251i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	911089.			0.767680			0.383840	−	0.923400i	$-0.374601\pi$
$269$	911089.			0.767680			0.383840	+	0.923400i	$0.374601\pi$
$270$		0			0
$271$	620129.			0.512931			0.256465	−	0.966553i	$-0.417442\pi$
$271$	620129.			0.512931			0.256465	+	0.966553i	$0.417442\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	21366.4	−	37007.8i	0.0170373	−	0.0295095i
$276$		0			0
$277$	118364.	+	205012.i	0.0926871	+	0.160539i	0.908641	−	0.417578i	$-0.137121\pi$
$277$	118364.	+	205012.i	0.0926871	+	0.160539i	−0.815954	+	0.578117i	$0.803788\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−538457.	−	932636.i	−0.406804	−	0.704606i	0.587725	−	0.809061i	$-0.300024\pi$
$281$	−538457.	−	932636.i	−0.406804	−	0.704606i	−0.994530	+	0.104455i	$0.966690\pi$
$282$		0			0
$283$	−1.02658e6	+	1.77809e6i	−0.761953	+	1.31974i	0.179890	+	0.983687i	$0.442426\pi$
$283$	−1.02658e6	+	1.77809e6i	−0.761953	+	1.31974i	−0.941843	+	0.336054i	$0.890908\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.59182e6			1.14074
$288$		0			0
$289$	3.09391e6			2.17903
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.06433e6	+	1.84347e6i	−0.724280	+	1.25449i	0.234990	+	0.971998i	$0.424494\pi$
$293$	−1.06433e6	+	1.84347e6i	−0.724280	+	1.25449i	−0.959270	+	0.282492i	$0.908839\pi$
$294$		0			0
$295$	596341.	+	1.03289e6i	0.398969	+	0.691035i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	931684.	+	1.61372e6i	0.602685	+	1.04388i
$300$		0			0
$301$	537780.	−	931463.i	0.342128	−	0.592583i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.62670e6			1.61682
$306$		0			0
$307$	−1.54154e6			−0.933489			−0.466744	−	0.884392i	$-0.654573\pi$
$307$	−1.54154e6			−0.933489			−0.466744	+	0.884392i	$0.654573\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	868197.	−	1.50376e6i	0.508999	−	0.881612i	−0.490946	−	0.871190i	$-0.663349\pi$
$311$	868197.	−	1.50376e6i	0.508999	−	0.881612i	0.999946	−	0.0104227i	$-0.00331772\pi$
$312$		0			0
$313$	−1.13707e6	−	1.96946e6i	−0.656034	−	1.13628i	−0.981634	−	0.190776i	$-0.938900\pi$
$313$	−1.13707e6	−	1.96946e6i	−0.656034	−	1.13628i	0.325600	−	0.945508i	$-0.394434\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	158418.	+	274388.i	0.0885435	+	0.153362i	0.906896	−	0.421355i	$-0.138445\pi$
$317$	158418.	+	274388.i	0.0885435	+	0.153362i	−0.818352	+	0.574717i	$0.805112\pi$
$318$		0			0
$319$	397038.	−	687690.i	0.218452	−	0.378369i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	63245.5			0.0337305
$324$		0			0
$325$	−115157.			−0.0604760
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.91774e6	+	3.32162e6i	−0.976787	+	1.69185i
$330$		0			0
$331$	224396.	+	388664.i	0.112576	+	0.194987i	0.916808	−	0.399328i	$-0.130757\pi$
$331$	224396.	+	388664.i	0.112576	+	0.194987i	−0.804232	+	0.594315i	$0.797423\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−550330.	−	953199.i	−0.267924	−	0.464057i
$336$		0			0
$337$	929347.	−	1.60968e6i	0.445762	−	0.772083i	−0.552343	−	0.833617i	$-0.686266\pi$
$337$	929347.	−	1.60968e6i	0.445762	−	0.772083i	0.998105	+	0.0615344i	$0.0195994\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.38051e6			−0.642914
$342$		0			0
$343$	−1.41216e6			−0.648111
$344$		0			0
$345$		0			0
$346$		0			0
$347$	111822.	−	193681.i	0.0498542	−	0.0863500i	−0.840021	−	0.542553i	$-0.817458\pi$
$347$	111822.	−	193681.i	0.0498542	−	0.0863500i	0.889876	+	0.456203i	$0.150791\pi$
$348$		0			0
$349$	−669904.	−	1.16031e6i	−0.294407	−	0.509929i	0.680439	−	0.732804i	$-0.261789\pi$
$349$	−669904.	−	1.16031e6i	−0.294407	−	0.509929i	−0.974847	+	0.222876i	$0.928456\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.69842e6	−	2.94174e6i	−0.725449	−	1.25652i	−0.958789	−	0.284120i	$-0.908299\pi$
$353$	−1.69842e6	−	2.94174e6i	−0.725449	−	1.25652i	0.233339	−	0.972395i	$-0.425035\pi$
$354$		0			0
$355$	−664307.	+	1.15061e6i	−0.279768	+	0.484572i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−761965.			−0.312032			−0.156016	−	0.987755i	$-0.549865\pi$
$359$	−761965.			−0.312032			−0.156016	+	0.987755i	$0.549865\pi$
$360$		0			0
$361$	−2.47521e6			−0.999642
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.58323e6	−	4.47428e6i	1.01492	−	1.75789i
$366$		0			0
$367$	−698558.	−	1.20994e6i	−0.270730	−	0.468919i	0.698319	−	0.715787i	$-0.253932\pi$
$367$	−698558.	−	1.20994e6i	−0.270730	−	0.468919i	−0.969049	+	0.246868i	$0.920599\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.99896e6	+	3.46229e6i	0.753995	+	1.30596i
$372$		0			0
$373$	−1.62621e6	+	2.81668e6i	−0.605209	+	1.04825i	0.386809	+	0.922160i	$0.373577\pi$
$373$	−1.62621e6	+	2.81668e6i	−0.605209	+	1.04825i	−0.992018	+	0.126093i	$0.959756\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.13989e6			−0.775421
$378$		0			0
$379$	−700188.			−0.250390			−0.125195	−	0.992132i	$-0.539956\pi$
$379$	−700188.			−0.250390			−0.125195	+	0.992132i	$0.539956\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.15361e6	−	1.99810e6i	0.401847	−	0.696019i	−0.592102	−	0.805863i	$-0.701702\pi$
$383$	1.15361e6	−	1.99810e6i	0.401847	−	0.696019i	0.993949	+	0.109844i	$0.0350352\pi$
$384$		0			0
$385$	985576.	+	1.70707e6i	0.338874	+	0.586947i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.00563e6	+	1.74180e6i	0.336950	+	0.583614i	0.983857	−	0.178955i	$-0.0572715\pi$
$389$	1.00563e6	+	1.74180e6i	0.336950	+	0.583614i	−0.646908	+	0.762568i	$0.723938\pi$
$390$		0			0
$391$	−3.36901e6	+	5.83530e6i	−1.11445	+	1.93028i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.96072e6			−0.632299
$396$		0			0
$397$	−165619.			−0.0527392			−0.0263696	−	0.999652i	$-0.508395\pi$
$397$	−165619.			−0.0527392			−0.0263696	+	0.999652i	$0.508395\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−547270.	+	947900.i	−0.169958	+	0.294375i	−0.938405	−	0.345538i	$-0.887696\pi$
$401$	−547270.	+	947900.i	−0.169958	+	0.294375i	0.768447	+	0.639913i	$0.221030\pi$
$402$		0			0
$403$	1.86011e6	+	3.22180e6i	0.570526	+	0.988180i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−919595.	−	1.59279e6i	−0.275176	−	0.476619i
$408$		0			0
$409$	536465.	−	929184.i	0.158574	−	0.274659i	−0.775780	−	0.631003i	$-0.782644\pi$
$409$	536465.	−	929184.i	0.158574	−	0.274659i	0.934355	+	0.356344i	$0.115977\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.24691e6			−0.936687
$414$		0			0
$415$	−6.25250e6			−1.78211
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−426090.	+	738010.i	−0.118568	+	0.205365i	−0.919200	−	0.393790i	$-0.871164\pi$
$419$	−426090.	+	738010.i	−0.118568	+	0.205365i	0.800633	+	0.599156i	$0.204497\pi$
$420$		0			0
$421$	−1.59392e6	−	2.76075e6i	−0.438290	−	0.759140i	0.559268	−	0.828987i	$-0.311082\pi$
$421$	−1.59392e6	−	2.76075e6i	−0.438290	−	0.759140i	−0.997558	+	0.0698468i	$0.977749\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−208207.	−	360625.i	−0.0559143	−	0.0968465i
$426$		0			0
$427$	−3.57541e6	+	6.19280e6i	−0.948979	+	1.64368i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.54409e6			−1.17829			−0.589147	−	0.808026i	$-0.700536\pi$
$431$	−4.54409e6			−1.17829			−0.589147	+	0.808026i	$0.700536\pi$
$432$		0			0
$433$	−6.16131e6			−1.57926			−0.789629	−	0.613585i	$-0.789727\pi$
$433$	−6.16131e6			−1.57926			−0.789629	+	0.613585i	$0.789727\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−47205.5	+	81762.4i	−0.0118247	+	0.0204809i
$438$		0			0
$439$	−451417.	−	781877.i	−0.111793	−	0.193632i	0.804700	−	0.593682i	$-0.202326\pi$
$439$	−451417.	−	781877.i	−0.111793	−	0.193632i	−0.916493	+	0.400050i	$0.868993\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.84356e6	+	3.19314e6i	0.446321	+	0.773051i	0.998143	−	0.0609110i	$-0.0194006\pi$
$443$	1.84356e6	+	3.19314e6i	0.446321	+	0.773051i	−0.551822	+	0.833962i	$0.686067\pi$
$444$		0			0
$445$	3.17113e6	−	5.49255e6i	0.759126	−	1.31484i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.71368e6			1.80570			0.902851	−	0.429954i	$-0.141470\pi$
$449$	7.71368e6			1.80570			0.902851	+	0.429954i	$0.141470\pi$
$450$		0			0
$451$	−2.21218e6			−0.512128
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.65595e6	−	4.60024e6i	0.601438	−	1.04172i
$456$		0			0
$457$	−1.89634e6	−	3.28455e6i	−0.424742	−	0.735674i	0.571654	−	0.820494i	$-0.306302\pi$
$457$	−1.89634e6	−	3.28455e6i	−0.424742	−	0.735674i	−0.996396	+	0.0848200i	$0.972968\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.38949e6	+	4.13871e6i	0.523664	+	0.907012i	0.999621	+	0.0275435i	$0.00876848\pi$
$461$	2.38949e6	+	4.13871e6i	0.523664	+	0.907012i	−0.475957	+	0.879469i	$0.657898\pi$
$462$		0			0
$463$	−1.27951e6	+	2.21618e6i	−0.277391	+	0.480456i	−0.970736	−	0.240151i	$-0.922803\pi$
$463$	−1.27951e6	+	2.21618e6i	−0.277391	+	0.480456i	0.693344	+	0.720606i	$0.256136\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.22563e6			1.74533			0.872664	−	0.488321i	$-0.162390\pi$
$467$	8.22563e6			1.74533			0.872664	+	0.488321i	$0.162390\pi$
$468$		0			0
$469$	2.99639e6			0.629022
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−747363.	+	1.29447e6i	−0.153596	+	0.266035i
$474$		0			0
$475$	−2917.33	−	5052.97i	−0.000593270	−	0.00102757i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.27184e6	−	2.20290e6i	−0.253277	−	0.438688i	0.711149	−	0.703041i	$-0.248175\pi$
$479$	−1.27184e6	−	2.20290e6i	−0.253277	−	0.438688i	−0.964426	+	0.264353i	$0.914842\pi$
$480$		0			0
$481$	−2.47814e6	+	4.29227e6i	−0.488386	+	0.845910i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.66153e6			−0.320741
$486$		0			0
$487$	1.25686e6			0.240139			0.120070	−	0.992765i	$-0.461688\pi$
$487$	1.25686e6			0.240139			0.120070	+	0.992765i	$0.461688\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4.94983e6	+	8.57336e6i	−0.926588	+	1.60490i	−0.137601	+	0.990488i	$0.543939\pi$
$491$	−4.94983e6	+	8.57336e6i	−0.926588	+	1.60490i	−0.788987	+	0.614410i	$0.789394\pi$
$492$		0			0
$493$	−3.86897e6	−	6.70125e6i	−0.716932	−	1.24176i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.80848e6	−	3.13238e6i	−0.328415	−	0.568831i
$498$		0			0
$499$	1.52076e6	−	2.63403e6i	0.273406	−	0.473553i	−0.696326	−	0.717726i	$-0.745183\pi$
$499$	1.52076e6	−	2.63403e6i	0.273406	−	0.473553i	0.969732	+	0.244173i	$0.0785164\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2.75985e6			0.486368			0.243184	−	0.969980i	$-0.421808\pi$
$503$	2.75985e6			0.486368			0.243184	+	0.969980i	$0.421808\pi$
$504$		0			0
$505$	−2.32186e6			−0.405142
$506$		0			0
$507$		0			0
$508$		0			0
$509$	722539.	−	1.25147e6i	0.123614	−	0.214105i	−0.797577	−	0.603218i	$-0.793885\pi$
$509$	722539.	−	1.25147e6i	0.123614	−	0.214105i	0.921190	+	0.389113i	$0.127218\pi$
$510$		0			0
$511$	7.03246e6	+	1.21806e7i	1.19139	+	2.06355i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.89384e6	−	1.02084e7i	−0.979220	−	1.69606i
$516$		0			0
$517$	2.66512e6	−	4.61612e6i	0.438521	−	0.759540i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.28492e6			0.207388			0.103694	−	0.994609i	$-0.466934\pi$
$521$	1.28492e6			0.207388			0.103694	+	0.994609i	$0.466934\pi$
$522$		0			0
$523$	−2.38275e6			−0.380912			−0.190456	−	0.981696i	$-0.560997\pi$
$523$	−2.38275e6			−0.380912			−0.190456	+	0.981696i	$0.560997\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.72623e6	+	1.16502e7i	−1.05498	+	1.82728i
$528$		0			0
$529$	−1.81100e6	−	3.13674e6i	−0.281370	−	0.487347i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.98070e6	+	5.16273e6i	0.454465	+	0.787157i
$534$		0			0
$535$	−4.72671e6	+	8.18691e6i	−0.713961	+	1.23662i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.70184e6			−0.252317
$540$		0			0
$541$	2.22158e6			0.326339			0.163170	−	0.986598i	$-0.447828\pi$
$541$	2.22158e6			0.326339			0.163170	+	0.986598i	$0.447828\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.79062e6	−	6.56554e6i	0.546662	−	0.946846i
$546$		0			0
$547$	908447.	+	1.57348e6i	0.129817	+	0.224850i	0.923606	−	0.383344i	$-0.125228\pi$
$547$	908447.	+	1.57348e6i	0.129817	+	0.224850i	−0.793789	+	0.608194i	$0.791894\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−54210.8	−	93895.8i	−0.00760688	−	0.0131755i
$552$		0			0
$553$	2.66889e6	−	4.62265e6i	0.371123	−	0.642803i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.53109e6			−0.482248			−0.241124	−	0.970494i	$-0.577516\pi$
$557$	−3.53109e6			−0.482248			−0.241124	+	0.970494i	$0.577516\pi$
$558$		0			0
$559$	4.02801e6			0.545207
$560$		0			0
$561$		0			0
$562$		0			0
$563$	163728.	−	283585.i	0.0217697	−	0.0377062i	−0.854935	−	0.518735i	$-0.826403\pi$
$563$	163728.	−	283585.i	0.0217697	−	0.0377062i	0.876705	+	0.481028i	$0.159737\pi$
$564$		0			0
$565$	2.71780e6	+	4.70737e6i	0.358176	+	0.620379i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.03386e6	−	6.98686e6i	−0.522325	−	0.904693i	−0.999663	−	0.0259731i	$-0.991732\pi$
$569$	−4.03386e6	−	6.98686e6i	−0.522325	−	0.904693i	0.477338	−	0.878720i	$-0.341602\pi$
$570$		0			0
$571$	105674.	−	183033.i	0.0135637	−	0.0234931i	−0.859164	−	0.511701i	$-0.829016\pi$
$571$	105674.	−	183033.i	0.0135637	−	0.0234931i	0.872728	+	0.488207i	$0.162349\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	621612.			0.0784060
$576$		0			0
$577$	−1.60363e6			−0.200523			−0.100261	−	0.994961i	$-0.531968\pi$
$577$	−1.60363e6			−0.200523			−0.100261	+	0.994961i	$0.531968\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.51078e6	−	1.47411e7i	1.04599	−	1.81171i
$582$		0			0
$583$	−2.77799e6	−	4.81161e6i	−0.338500	−	0.586299i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.96240e6	−	8.59513e6i	−0.594424	−	1.02957i	−0.993628	−	0.112711i	$-0.964047\pi$
$587$	−4.96240e6	−	8.59513e6i	−0.594424	−	1.02957i	0.399203	−	0.916862i	$-0.369287\pi$
$588$		0			0
$589$	−94245.9	+	163239.i	−0.0111937	+	0.0193881i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.74316e6			0.670678			0.335339	−	0.942098i	$-0.391149\pi$
$593$	5.74316e6			0.670678			0.335339	+	0.942098i	$0.391149\pi$
$594$		0			0
$595$	1.92081e7			2.22429
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.63787e6	+	4.56892e6i	−0.300391	+	0.520292i	−0.976224	−	0.216762i	$-0.930450\pi$
$599$	−2.63787e6	+	4.56892e6i	−0.300391	+	0.520292i	0.675834	+	0.737054i	$0.263784\pi$
$600$		0			0
$601$	−2.07476e6	−	3.59359e6i	−0.234305	−	0.405828i	0.724765	−	0.688996i	$-0.241948\pi$
$601$	−2.07476e6	−	3.59359e6i	−0.234305	−	0.405828i	−0.959070	+	0.283167i	$0.908615\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.27086e6	+	5.66530e6i	0.363307	+	0.629266i
$606$		0			0
$607$	7.92504e6	−	1.37266e7i	0.873031	−	1.51213i	0.0141852	−	0.999899i	$-0.495485\pi$
$607$	7.92504e6	−	1.37266e7i	0.873031	−	1.51213i	0.858846	−	0.512234i	$-0.171182\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.43640e7			−1.55658
$612$		0			0
$613$	1.48963e7			1.60113			0.800565	−	0.599246i	$-0.204533\pi$
$613$	1.48963e7			1.60113			0.800565	+	0.599246i	$0.204533\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.23952e6	+	1.25392e7i	−0.765591	+	1.32604i	0.174343	+	0.984685i	$0.444220\pi$
$617$	−7.23952e6	+	1.25392e7i	−0.765591	+	1.32604i	−0.939934	+	0.341357i	$0.889113\pi$
$618$		0			0
$619$	5.29461e6	+	9.17053e6i	0.555402	+	0.961984i	0.997872	+	0.0652007i	$0.0207688\pi$
$619$	5.29461e6	+	9.17053e6i	0.555402	+	0.961984i	−0.442471	+	0.896783i	$0.645898\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.63294e6	+	1.49527e7i	0.891125	+	1.54347i
$624$		0			0
$625$	5.16985e6	−	8.95445e6i	0.529393	−	0.916936i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.79221e7			−1.80619
$630$		0			0
$631$	−4.93118e6			−0.493035			−0.246517	−	0.969138i	$-0.579286\pi$
$631$	−4.93118e6			−0.493035			−0.246517	+	0.969138i	$0.579286\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.73480e6	−	1.51291e7i	0.859644	−	1.48895i
$636$		0			0
$637$	2.29307e6	+	3.97171e6i	0.223908	+	0.387819i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.50299e6	+	1.64597e7i	0.913514	+	1.58225i	0.809062	+	0.587723i	$0.199975\pi$
$641$	9.50299e6	+	1.64597e7i	0.913514	+	1.58225i	0.104452	+	0.994530i	$0.466691\pi$
$642$		0			0
$643$	3.34869e6	−	5.80010e6i	0.319409	−	0.553233i	−0.660956	−	0.750425i	$-0.729849\pi$
$643$	3.34869e6	−	5.80010e6i	0.319409	−	0.553233i	0.980365	+	0.197192i	$0.0631822\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.62821e7			1.52915			0.764574	−	0.644536i	$-0.222949\pi$
$647$	1.62821e7			1.52915			0.764574	+	0.644536i	$0.222949\pi$
$648$		0			0
$649$	4.51228e6			0.420518
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.76043e6	−	4.78121e6i	0.253334	−	0.438788i	−0.711108	−	0.703083i	$-0.751806\pi$
$653$	2.76043e6	−	4.78121e6i	0.253334	−	0.438788i	0.964442	+	0.264296i	$0.0851394\pi$
$654$		0			0
$655$	−9.12175e6	−	1.57993e7i	−0.830759	−	1.43892i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−69588.1	−	120530.i	−0.00624197	−	0.0108114i	0.862888	−	0.505396i	$-0.168654\pi$
$659$	−69588.1	−	120530.i	−0.00624197	−	0.0108114i	−0.869130	+	0.494585i	$0.835320\pi$
$660$		0			0
$661$	9.72751e6	−	1.68485e7i	0.865960	−	1.49989i	−0.000129981	−	1.00000i	$-0.500041\pi$
$661$	9.72751e6	−	1.68485e7i	0.865960	−	1.49989i	0.866090	−	0.499887i	$-0.166625\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	269137.			0.0236004
$666$		0			0
$667$	1.15510e7			1.00532
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.96882e6	−	8.60624e6i	0.426036	−	0.737916i
$672$		0			0
$673$	4.57506e6	+	7.92423e6i	0.389367	+	0.674403i	0.992364	−	0.123340i	$-0.0393605\pi$
$673$	4.57506e6	+	7.92423e6i	0.389367	+	0.674403i	−0.602998	+	0.797743i	$0.706027\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.96358e6	+	8.59717e6i	0.416220	+	0.720915i	0.995556	−	0.0941744i	$-0.0300211\pi$
$677$	4.96358e6	+	8.59717e6i	0.416220	+	0.720915i	−0.579335	+	0.815089i	$0.696688\pi$
$678$		0			0
$679$	2.26164e6	−	3.91728e6i	0.188256	−	0.326069i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.27969e7			1.04967			0.524835	−	0.851204i	$-0.324127\pi$
$683$	1.27969e7			1.04967			0.524835	+	0.851204i	$0.324127\pi$
$684$		0			0
$685$	−56646.0			−0.00461257
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.48616e6	+	1.29664e7i	−0.600774	+	1.04057i
$690$		0			0
$691$	−4.02846e6	−	6.97749e6i	−0.320955	−	0.555910i	0.659731	−	0.751502i	$-0.270670\pi$
$691$	−4.02846e6	−	6.97749e6i	−0.320955	−	0.555910i	−0.980685	+	0.195592i	$0.937337\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.23406e6	−	3.86951e6i	−0.175442	−	0.303874i
$696$		0			0
$697$	−1.07784e7	+	1.86687e7i	−0.840371	+	1.45556i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.05658e7			−0.812100			−0.406050	−	0.913851i	$-0.633094\pi$
$701$	−1.05658e7			−0.812100			−0.406050	+	0.913851i	$0.633094\pi$
$702$		0			0
$703$	−251120.			−0.0191643
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.16046e6	−	5.47408e6i	0.237795	−	0.411872i
$708$		0			0
$709$	7.59109e6	+	1.31482e7i	0.567138	+	0.982312i	0.996847	+	0.0793445i	$0.0252827\pi$
$709$	7.59109e6	+	1.31482e7i	0.567138	+	0.982312i	−0.429709	+	0.902967i	$0.641384\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.00407e7	−	1.73911e7i	−0.739676	−	1.28116i
$714$		0			0
$715$	−3.69102e6	+	6.39303e6i	−0.270011	+	0.467672i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.93525e6			0.211750			0.105875	−	0.994379i	$-0.466236\pi$
$719$	2.93525e6			0.211750			0.105875	+	0.994379i	$0.466236\pi$
$720$		0			0
$721$	3.20903e7			2.29898
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−356929.	+	618219.i	−0.0252195	+	0.0436815i
$726$		0			0
$727$	4.66682e6	+	8.08316e6i	0.327480	+	0.567212i	0.982011	−	0.188823i	$-0.0604673\pi$
$727$	4.66682e6	+	8.08316e6i	0.327480	+	0.567212i	−0.654531	+	0.756035i	$0.727134\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7.28274e6	+	1.26141e7i	0.504082	+	0.873096i
$732$		0			0
$733$	2.36625e6	−	4.09846e6i	0.162667	−	0.281748i	−0.773157	−	0.634215i	$-0.781324\pi$
$733$	2.36625e6	−	4.09846e6i	0.162667	−	0.281748i	0.935824	+	0.352466i	$0.114657\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.16413e6			−0.282394
$738$		0			0
$739$	−2.42162e7			−1.63115			−0.815575	−	0.578651i	$-0.803579\pi$
$739$	−2.42162e7			−1.63115			−0.815575	+	0.578651i	$0.803579\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.23724e7	+	2.14296e7i	−0.822207	+	1.42410i	0.0818282	+	0.996646i	$0.473924\pi$
$743$	−1.23724e7	+	2.14296e7i	−0.822207	+	1.42410i	−0.904035	+	0.427458i	$0.859409\pi$
$744$		0			0
$745$	−3.43490e6	−	5.94943e6i	−0.226738	−	0.392721i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.28678e7	−	2.22877e7i	−0.838108	−	1.45165i
$750$		0			0
$751$	2.78532e6	−	4.82432e6i	0.180209	−	0.312131i	−0.761743	−	0.647880i	$-0.775656\pi$
$751$	2.78532e6	−	4.82432e6i	0.180209	−	0.312131i	0.941952	+	0.335749i	$0.108989\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.43284e6			−0.219173
$756$		0			0
$757$	−2.72203e6			−0.172645			−0.0863224	−	0.996267i	$-0.527511\pi$
$757$	−2.72203e6			−0.172645			−0.0863224	+	0.996267i	$0.527511\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.53087e6	−	1.30438e7i	0.471393	−	0.816477i	−0.528071	−	0.849200i	$-0.677085\pi$
$761$	7.53087e6	−	1.30438e7i	0.471393	−	0.816477i	0.999464	+	0.0327230i	$0.0104179\pi$
$762$		0			0
$763$	1.03194e7	+	1.78738e7i	0.641718	+	1.11149i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.07989e6	−	1.05307e7i	−0.373170	−	0.646350i
$768$		0			0
$769$	−4.56946e6	+	7.91454e6i	−0.278644	+	0.482625i	−0.971048	−	0.238885i	$-0.923218\pi$
$769$	−4.56946e6	+	7.91454e6i	−0.278644	+	0.482625i	0.692404	+	0.721510i	$0.256552\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.35738e7			0.817060			0.408530	−	0.912745i	$-0.366041\pi$
$773$	1.35738e7			0.817060			0.408530	+	0.912745i	$0.366041\pi$
$774$		0			0
$775$	1.24105e6			0.0742223
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−151023.	+	261580.i	−0.00891661	+	0.0154440i
$780$		0			0
$781$	2.51328e6	+	4.35312e6i	0.147439	+	0.255372i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8.57395e6	−	1.48505e7i	−0.496600	−	0.860136i
$786$		0			0
$787$	1.02146e7	−	1.76922e7i	0.587875	−	1.01823i	−0.406635	−	0.913591i	$-0.633298\pi$
$787$	1.02146e7	−	1.76922e7i	0.587875	−	1.01823i	0.994510	−	0.104639i	$-0.0333686\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.47976e7			−0.840913
$792$		0			0
$793$	−2.67801e7			−1.51227
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.51983e7	+	2.63242e7i	−0.847518	+	1.46794i	0.0358980	+	0.999355i	$0.488571\pi$
$797$	−1.51983e7	+	2.63242e7i	−0.847518	+	1.46794i	−0.883416	+	0.468589i	$0.844762\pi$
$798$		0			0
$799$	−2.59705e7	−	4.49822e7i	−1.43917	−	2.49272i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9.77314e6	−	1.69276e7i	−0.534866	−	0.926416i
$804$		0			0
$805$	−1.43366e7	+	2.48318e7i	−0.779754	+	1.35057i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.50428e7			1.88247			0.941234	−	0.337755i	$-0.109667\pi$
$809$	3.50428e7			1.88247			0.941234	+	0.337755i	$0.109667\pi$
$810$		0			0
$811$	1.55069e7			0.827893			0.413946	−	0.910301i	$-0.364150\pi$
$811$	1.55069e7			0.827893			0.413946	+	0.910301i	$0.364150\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.84343e6	−	4.92496e6i	0.149951	−	0.259722i
$816$		0			0
$817$	102043.	+	176745.i	0.00534848	+	0.00926383i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.58087e6	+	1.31304e7i	0.392519	+	0.679863i	0.992781	−	0.119940i	$-0.0382703\pi$
$821$	7.58087e6	+	1.31304e7i	0.392519	+	0.679863i	−0.600262	+	0.799804i	$0.704937\pi$
$822$		0			0
$823$	6.32661e6	−	1.09580e7i	0.325590	−	0.563939i	−0.656041	−	0.754725i	$-0.727770\pi$
$823$	6.32661e6	−	1.09580e7i	0.325590	−	0.563939i	0.981632	+	0.190786i	$0.0611037\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.21240e7			0.616427			0.308213	−	0.951317i	$-0.400269\pi$
$827$	1.21240e7			0.616427			0.308213	+	0.951317i	$0.400269\pi$
$828$		0			0
$829$	−1.79003e7			−0.904634			−0.452317	−	0.891857i	$-0.649402\pi$
$829$	−1.79003e7			−0.904634			−0.452317	+	0.891857i	$0.649402\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8.29185e6	+	1.43619e7i	−0.414037	+	0.717133i
$834$		0			0
$835$	534588.	+	925933.i	0.0265340	+	0.0459583i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.17183e7	+	2.02966e7i	0.574723	+	0.995449i	0.996072	+	0.0885507i	$0.0282235\pi$
$839$	1.17183e7	+	2.02966e7i	0.574723	+	0.995449i	−0.421349	+	0.906899i	$0.638443\pi$
$840$		0			0
$841$	3.62301e6	−	6.27524e6i	0.176636	−	0.305943i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.50368e6			−0.0724458
$846$		0			0
$847$	−1.78089e7			−0.852960
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.33768e7	−	2.31694e7i	0.633183	−	1.09671i
$852$		0			0
$853$	−1.44546e7	−	2.50361e7i	−0.680196	−	1.17813i	−0.974921	−	0.222552i	$-0.928561\pi$
$853$	−1.44546e7	−	2.50361e7i	−0.680196	−	1.17813i	0.294725	−	0.955582i	$-0.404772\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5.34804e6	−	9.26307e6i	−0.248738	−	0.430827i	0.714438	−	0.699699i	$-0.246682\pi$
$857$	−5.34804e6	−	9.26307e6i	−0.248738	−	0.430827i	−0.963176	+	0.268872i	$0.913349\pi$
$858$		0			0
$859$	−1.84851e7	+	3.20172e7i	−0.854752	+	1.48047i	0.0221237	+	0.999755i	$0.492957\pi$
$859$	−1.84851e7	+	3.20172e7i	−0.854752	+	1.48047i	−0.876875	+	0.480718i	$0.840376\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.83720e7			−1.75383			−0.876914	−	0.480646i	$-0.840402\pi$
$863$	−3.83720e7			−1.75383			−0.876914	+	0.480646i	$0.840402\pi$
$864$		0			0
$865$	−2.86211e7			−1.30061
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.70900e6	+	6.42417e6i	−0.166612	+	0.288581i
$870$		0			0
$871$	5.61079e6	+	9.71817e6i	0.250599	+	0.434050i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.32405e7	+	2.29332e7i	0.584633	+	1.01261i
$876$		0			0
$877$	−1.83104e7	+	3.17146e7i	−0.803895	+	1.39239i	0.113140	+	0.993579i	$0.463909\pi$
$877$	−1.83104e7	+	3.17146e7i	−0.803895	+	1.39239i	−0.917035	+	0.398808i	$0.869424\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.70223e7			−0.738889			−0.369445	−	0.929253i	$-0.620452\pi$
$881$	−1.70223e7			−0.738889			−0.369445	+	0.929253i	$0.620452\pi$
$882$		0			0
$883$	3.54409e7			1.52969			0.764844	−	0.644215i	$-0.222816\pi$
$883$	3.54409e7			1.52969			0.764844	+	0.644215i	$0.222816\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.00360e6	−	1.03985e7i	0.256214	−	0.443776i	−0.709011	−	0.705198i	$-0.750858\pi$
$887$	6.00360e6	−	1.03985e7i	0.256214	−	0.443776i	0.965225	+	0.261422i	$0.0841915\pi$
$888$		0			0
$889$	2.37793e7	+	4.11869e7i	1.00912	+	1.74785i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−363890.	−	630276.i	−0.0152701	−	0.0264486i
$894$		0			0
$895$	4.32418e6	−	7.48971e6i	0.180446	−	0.312541i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.30615e7			0.951676
$900$		0			0
$901$	−5.41406e7			−2.22183
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.11828e6	−	3.66898e6i	0.0859731	−	0.148910i
$906$		0			0
$907$	4.21452e6	+	7.29977e6i	0.170110	+	0.294639i	0.938458	−	0.345393i	$-0.112254\pi$
$907$	4.21452e6	+	7.29977e6i	0.170110	+	0.294639i	−0.768348	+	0.640032i	$0.778921\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.24424e7	+	2.15508e7i	0.496715	+	0.860335i	0.999993	−	0.00378921i	$-0.00120614\pi$
$911$	1.24424e7	+	2.15508e7i	0.496715	+	0.860335i	−0.503278	+	0.864125i	$0.667873\pi$
$912$		0			0
$913$	−1.18276e7	+	2.04860e7i	−0.469590	+	0.813354i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.96653e7			1.95043
$918$		0			0
$919$	−3.68789e7			−1.44042			−0.720210	−	0.693757i	$-0.755954\pi$
$919$	−3.68789e7			−1.44042			−0.720210	+	0.693757i	$0.755954\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.77282e6	−	1.17309e7i	0.261677	−	0.453238i
$924$		0			0
$925$	826697.	+	1.43188e6i	0.0317682	+	0.0550241i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.30455e7	+	2.25955e7i	0.495932	+	0.858979i	0.999989	−	0.00469142i	$-0.00149333\pi$
$929$	1.30455e7	+	2.25955e7i	0.495932	+	0.858979i	−0.504057	+	0.863670i	$0.668160\pi$
$930$		0			0
$931$	−116183.	+	201235.i	−0.00439307	+	0.00760901i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.66938e7			−0.998576
$936$		0			0
$937$	7.84259e6			0.291817			0.145908	−	0.989298i	$-0.453390\pi$
$937$	7.84259e6			0.291817			0.145908	+	0.989298i	$0.453390\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.68082e7	−	2.91127e7i	0.618796	−	1.07179i	−0.370910	−	0.928669i	$-0.620954\pi$
$941$	1.68082e7	−	2.91127e7i	0.618796	−	1.07179i	0.989706	−	0.143117i	$-0.0457126\pi$
$942$		0			0
$943$	−1.60896e7	−	2.78681e7i	−0.589206	−	1.02053i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.88767e6	−	1.36619e7i	−0.285808	−	0.495034i	0.686997	−	0.726660i	$-0.258929\pi$
$947$	−7.88767e6	−	1.36619e7i	−0.285808	−	0.495034i	−0.972805	+	0.231627i	$0.925595\pi$
$948$		0			0
$949$	−2.63368e7	+	4.56167e7i	−0.949288	+	1.64421i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.32737e7			−0.830104			−0.415052	−	0.909798i	$-0.636237\pi$
$953$	−2.32737e7			−0.830104			−0.415052	+	0.909798i	$0.636237\pi$
$954$		0			0
$955$	−3.45376e7			−1.22542
$956$		0			0
$957$		0			0
$958$		0			0
$959$	77105.3	−	133550.i	0.00270731	−	0.00468920i
$960$		0			0
$961$	−5.73176e6	−	9.92770e6i	−0.200207	−	0.346769i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.08541e7	+	3.61203e7i	0.720896	+	1.24863i
$966$		0			0
$967$	−9.77148e6	+	1.69247e7i	−0.336043	+	0.582043i	−0.983685	−	0.179902i	$-0.942422\pi$
$967$	−9.77148e6	+	1.69247e7i	−0.336043	+	0.582043i	0.647642	+	0.761945i	$0.275755\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5.59842e7			−1.90554			−0.952768	−	0.303700i	$-0.901778\pi$
$971$	−5.59842e7			−1.90554			−0.952768	+	0.303700i	$0.901778\pi$
$972$		0			0
$973$	1.21638e7			0.411897
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.10112e6	+	3.63925e6i	−0.0704231	+	0.121976i	−0.899087	−	0.437770i	$-0.855768\pi$
$977$	−2.10112e6	+	3.63925e6i	−0.0704231	+	0.121976i	0.828664	+	0.559747i	$0.189102\pi$
$978$		0			0
$979$	−1.19974e7	−	2.07800e7i	−0.400063	−	0.692930i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−9.07877e6	−	1.57249e7i	−0.299670	−	0.519044i	0.676390	−	0.736543i	$-0.263543\pi$
$983$	−9.07877e6	−	1.57249e7i	−0.299670	−	0.519044i	−0.976060	+	0.217500i	$0.930210\pi$
$984$		0			0
$985$	1.20036e7	−	2.07909e7i	0.394205	−	0.682782i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.17429e7			−0.706851
$990$		0			0
$991$	6.51450e6			0.210716			0.105358	−	0.994434i	$-0.466401\pi$
$991$	6.51450e6			0.210716			0.105358	+	0.994434i	$0.466401\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.98292e7	+	3.43451e7i	−0.634960	+	1.09978i
$996$		0			0
$997$	1.44585e7	+	2.50428e7i	0.460664	+	0.797894i	0.998994	−	0.0448401i	$-0.0142778\pi$
$997$	1.44585e7	+	2.50428e7i	0.460664	+	0.797894i	−0.538330	+	0.842734i	$0.680944\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.6.e.f.109.1		4
3.2	odd	2		324.6.e.g.109.2		4
9.2	odd	6		324.6.e.g.217.2		4
9.4	even	3		108.6.a.c.1.2	yes	2
9.5	odd	6		108.6.a.b.1.1	✓	2
9.7	even	3	inner	324.6.e.f.217.1		4
36.23	even	6		432.6.a.r.1.1		2
36.31	odd	6		432.6.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.a.b.1.1	✓	2	9.5	odd	6
108.6.a.c.1.2	yes	2	9.4	even	3
324.6.e.f.109.1		4	1.1	even	1	trivial
324.6.e.f.217.1		4	9.7	even	3	inner
324.6.e.g.109.2		4	3.2	odd	2
324.6.e.g.217.2		4	9.2	odd	6
432.6.a.q.1.2		2	36.31	odd	6
432.6.a.r.1.1		2	36.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-28.8141 + 49.9074i) q^{5} +(-78.4422 - 135.866i) q^{7} +O(q^{10})\)	\(q+(-28.8141 + 49.9074i) q^{5} +(-78.4422 - 135.866i) q^{7} +(109.012 + 188.815i) q^{11} +(293.769 - 508.822i) q^{13} +2124.56 q^{17} +29.7687 q^{19} +(-1585.74 + 2746.59i) q^{23} +(-98.0000 - 169.741i) q^{25} +(-1821.07 - 3154.18i) q^{29} +(-3165.94 + 5483.57i) q^{31} +9040.95 q^{35} -8435.69 q^{37} +(-5073.22 + 8787.07i) q^{41} +(3427.88 + 5937.26i) q^{43} +(-12223.9 - 21172.4i) q^{47} +(-3902.85 + 6759.94i) q^{49} -25483.2 q^{53} -12564.4 q^{55} +(10348.1 - 17923.4i) q^{59} +(-22790.1 - 39473.7i) q^{61} +(16929.3 + 29322.5i) q^{65} +(-9549.67 + 16540.5i) q^{67} +23055.0 q^{71} -89651.6 q^{73} +(17102.4 - 29622.1i) q^{77} +(17011.8 + 29465.3i) q^{79} +(54248.7 + 93961.6i) q^{83} +(-61217.3 + 106031. i) q^{85} -110055. q^{89} -92175.4 q^{91} +(-857.757 + 1485.68i) q^{95} +(14416.0 + 24969.2i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{7}+O(q^{10}) \) 4 * q + 32 * q^7	\( 4 q + 32 q^{7} - 486 q^{11} - 208 q^{13} + 3888 q^{17} - 1264 q^{19} - 6804 q^{23} - 392 q^{25} - 11664 q^{29} - 3328 q^{31} + 39852 q^{35} - 19912 q^{37} - 13608 q^{41} - 4960 q^{43} - 18468 q^{47} - 26676 q^{49} - 23328 q^{53} - 106272 q^{55} - 1944 q^{59} - 8176 q^{61} + 79704 q^{65} - 90064 q^{67} - 89424 q^{71} - 242428 q^{73} + 167184 q^{77} - 28768 q^{79} + 15066 q^{83} - 132840 q^{85} - 357696 q^{89} - 484880 q^{91} - 39852 q^{95} - 88942 q^{97}+O(q^{100}) \) 4 * q + 32 * q^7 - 486 * q^11 - 208 * q^13 + 3888 * q^17 - 1264 * q^19 - 6804 * q^23 - 392 * q^25 - 11664 * q^29 - 3328 * q^31 + 39852 * q^35 - 19912 * q^37 - 13608 * q^41 - 4960 * q^43 - 18468 * q^47 - 26676 * q^49 - 23328 * q^53 - 106272 * q^55 - 1944 * q^59 - 8176 * q^61 + 79704 * q^65 - 90064 * q^67 - 89424 * q^71 - 242428 * q^73 + 167184 * q^77 - 28768 * q^79 + 15066 * q^83 - 132840 * q^85 - 357696 * q^89 - 484880 * q^91 - 39852 * q^95 - 88942 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more